# Compton Scattering - Energy Conservation

1. Oct 22, 2009

### tramar

1. The problem statement, all variables and given/known data

Using conservation of relativistic energy and momentum, show that for Compton scattering from a free electron, the energy-angle formula for the electron is

$$T=m_0c^2\frac{2\alpha^2cos^2\phi}{1+2\alpha+\alpha^2sin^2\phi}$$

where

T = kinetic energy given to electon
$$\phi$$ = electron recoil angle
$$\alpha=\frac{h\nu}{m_0c^2}$$
$$h\nu$$ = incident photon energy
$$m_0c^2$$ = electron rest energy

2. Relevant equations
$$T=h\nu-h\nu'$$
$$h\nu+m_0c^2=h\nu'+\sqrt{(m_0c^2)^2+(cp_e)^2}$$
$$p_e^2=p_\gamma^2+p_{\gamma'}^2-2p_\gamma p_{\gamma'}}cos\phi$$

3. The attempt at a solution
I did a problem similar to this in which I proved that the photoelectric effect is impossible for a free electron. I've tried taking the expression for p and putting it into the energy conservation equation, but what is really baffling me is where is that sin^2 term coming from?? If someone can tell me where it comes in I will probably be well on my way to solving this problem... Thanks!